Lagrange’s four-square theorem asserts that each $ n\in\mathbb N=\{0,1,2,\ldots\}$ is the sum of four integer squares. This nice result is actually very weak, for example, it is a consequence of the Gauss-Legendre theorem on sums of three squares. As $ $ 4(w^2+x^2+y^2+z^2)=(2w)^2+(2x)^2+(2y)^2+(2z)^2,$ $ Lagrange’s four-square theorem is equivalent to that any positive integer not divisible by $ 4$ can be written as the sum of four squares.

Here I propose a new refinement of Lagrange’s four-square theorem.

QUESTION. Is my following conjecture true?

**Big 1-3-5 Conjecture**. Any positive integer $ n\not\equiv0\pmod8$ can be written as $ $ w^2+\left(\frac{x(x+1)}2\right)^2+\left(\frac{y(3y+1)}2\right)^2+\left(\frac{z(5z+1)}2\right)^2,$ $ where $ w$ is a positive integer and $ x,y,z$ are integers.

I have verified this for $ n$ up to $ 2\times10^7$ . For example, $ 28$ has a unique required representation: $ $ 28=3^2+\left(\frac {2(2+1)}2\right)^2+\left(\frac{(-1)(3\times(-1)+1)}2\right)^2+\left(\frac{1(5\times1+1)}2\right)^2.$ $ See also http://oeis.org/A306614 for related data and similar conjectures.

Note that the big 1-3-5 conjecture is different from my 1-3-5 conjecture published in this paper which states that any $ n\in\mathbb N$ can be written as $ x^2+y^2+z^2+w^2$ with $ x,y,z,w\in\mathbb N$ such that $ x+3y+5z$ is a square. We should also not confuse it with my little 1-3-5 conjecture (cf. http://oeis.org/A287616) which states that any $ n\in\mathbb N$ can be written as $ x(x+1)/2+y(3y+1)/2+z(5z+1)/2$ with $ x,y,z\in\mathbb N$ .

The Big 1-3-5 Conjecture is much stronger than Lagrange’s four-square theorem. Your comments or further check are welcome!